34edo
34edo is the equal tuning system which splits the octave into 34 equal steps, of about (1200/34) ~= 35.3 cents each. It is a 5-limit and 2.3.5.13 system with a number of melodically intuitive structures.
Derivation
From doubling 17edo
One can observe that 17edo's step is a nearly perfectly tuned 25/24, and also that 5/4 and 6/5 are almost exactly halfway in-between notes of 17edo. Thus, 17edo can be doubled to improve the tunings of 5-limit intervals.
From the usage of Pythagorean diatonic semitones as classical chromatic semitones
Forcing 17edo's near-just 25/24, which is a Pythagorean diatonic semitone, to surround a neutral third and function as a chromatic semitone, requires offsetting the chain of fifths by a perfect semioctave, effectively allowing one to 'swap' the tunings of diatonic and chromatic semitones. This results in the 34edo tuning of diaschismic.
From the DKW step sizes
DKW theory suggests that the core step sizes of the 5-limit are 9/8, 16/15, and 25/24. It can be observed that 16/15 stacks twice to approximate 9/8, and that 25/24 stacks 3 times to approximate 9/8. Tempering these equivalences together results in 34edo. Because the latter (kleismic) equalizes 24:25:26:27, and the former (diaschismic) equalizes 15:16:17:18, 34edo can be seen as a 2.3.5.13.17 system. 34edo can, thus, be broken up as 6-3-2-3-6-3-2-3-6, with 6 representing 9/8, 3 representing 16/15, and 2 representing 25/24.
Theory
Edostep interpretations
34edo's edostep, the sextula, has the following interpretations in the 2.3.5.13.17 subgroup:
- 81/80, the difference between the fifth-generated major third and the classical major third
- 128/125, the difference between the 5-limit enharmonic intervals
TODO: add to list
JI approximation
34edo is straddle-7, straddle-11, and straddle-19, but has good accuracy on the 2.3.5.13.17.23 subgroup. 34edo inherits 17edo's mosdiatonic scale, 6-6-2-6-6-6-2, with the "optimally tuned" leading tone approximating 25/24. It also supports the zarlino scale, but because it does not support porcupine, the zarlino scale requires 2 sets of accidentals to notate, making it awkward to use as the basis of notation. (The best option is to use 5-sharp and 5-flat accidentals from 17edo's diatonic, as if you are notating 5-limit JI, which may in 34edo be represented as ups and downs.)
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +3.9 | +1.9 | -15.9 | +13.4 | +6.5 | +0.9 | -15.2 | +7.0 | -6.0 | -15.6 |
| Relative (%) | 0.0 | +11.1 | +5.4 | -45.0 | +37.9 | +18.5 | +2.6 | -43.0 | +19.9 | -17.1 | -44.3 | |
| Steps
(reduced) |
34
(0) |
54
(20) |
79
(11) |
95
(27) |
118
(16) |
126
(24) |
139
(3) |
144
(8) |
154
(18) |
165
(29) |
168
(32) | |
| Quality | Inframinor | Farminor | Nearminor | Neutral | Nearmajor | Farmajor | Ultramajor |
|---|---|---|---|---|---|---|---|
| Cents | 247 | 282 | 318 | 353 | 388 | 424 | 459 |
| Just interpretation | 15/13 | 32/27 | 6/5 | 16/13 | 5/4 | 32/25 | 13/10 |
MOS diatonic thirds are bolded.
Chords
34edo supports arto and tendo theory with its inframinor and ultramajor thirds. Being a diaschismic edo, it has a series of tetrads wherein the third and seventh are separated by 600 cents, but due to not supporting pajara, these do not approximate simple 7-limit chords.
Scales
34edo contains 17edo's diatonic scale and alongside it a zarlino scale. Other scales it includes are:
- diaschismic[12], with steps 3-3-3-3-3-2-3-3-3-3-3-2 (LLLLLsLLLLLs)
- the Delkian scale, a MODMOS of diaschismic[12] with steps 3-3-3-2-3-3-3-3-3-3-2-3 (LLLsLLLLLLsL)
- the hemipythagorean decatonic, with steps 3-4-3-4-3-3-4-3-4-3 (sLsLssLsLs)
- the Sixanian scale, a MODMOS of the above with steps 3-4-3-4-3-3-3-4-3-4 (sLsLsssLsL)
- the Roklotian scale, 2-2-2-3-2-3-2-2-2-3-2-3-2-2-2 (sssLsLsssLsLsss)
- the MOS pentatonic, pythagorean[5], 6-8-6-8-6 (sLsLs)
- the equable pentatonic, semaphore[5], 7-7-6-7-7 (LLsLL)
- the vertical pentatonic, 5-9-6-5-9 (sLmsL)
Additional regular temperaments
Alongside kleismic (shared with 15edo and 19edo), and diaschismic (shared with 12edo), 34edo supports the following temperaments:
- Tetracot (splitting 3/2 into four 10/9s), shared with 7edo and 27edo)
- Gammic (setting 25/24 to a tenth of 3/2), shared with 103edo
68edo
68edo is the double of 34edo, and improves its mapping of 7 much as 34edo improves 17edo's mapping of 5. This improves the mappings of 11 and 19 as well, making 68edo function as a general 19-limit system.
The new 7/4 supports sensamagic, doubling 9/7 to reach 5/3, and 2.5.7 didacus, splitting 5/4 into two wholetones that stack 5 times to reach 7/4. Additionally, the new 11/8 makes 14/11 equal to 81/64, supporting pentacircle (and various gentle/neogothic temperaments).
